Temporal Casimir Rigidity and Closure of the Effective Metric: Identification of A_τ = 2

Overview

The Q5b paper left open problem O3: the identification of the temporal coefficient $A_\tau$ entering the effective Lorentzian co-metric. The spatial coefficients $A_H = A_Z = 2$ were established by Q8 and Q10. Q11 supplies the missing temporal identification.

The key observation is that the cascade increment operator — which advances BFS shells in the Weil representation — internalises no new irreducible representation (Schur rigidity). This absence of new irreps forces asymptotic su(2)-equivariance of the temporal sector, which in turn fixes $A_\tau = 2$, identical to the spatial Casimir value.

The full effective Lorentzian co-metric thereby reads \[ g^{\mu\nu} = \mathrm{diag}(-2,\; 2,\; 2,\; 2) \propto \eta^{\mu\nu}, \] establishing Minkowski signature from purely spectral admissibility data.

Status. Structurally motivated.

Core contributions

Schur rigidity of the cascade increment: the operator advancing BFS shells has no new irreducible decomposition component, so no new su(2)-representation is introduced at each cascade step.
Asymptotic su(2)-equivariance of the temporal sector: Schur rigidity propagates su(2)-equivariance from the spatial sector (Q8/Q10) to the temporal sector, constraining the temporal quadratic form to be Casimir-valued.
Identification $A_\tau = 2$: the unique su(2)-equivariant quadratic form for the temporal degree of freedom takes the value 2, matching the spatial Casimir.
Full Minkowski co-metric: combining $A_\tau = 2$ (Q11) with $A_H = A_Z = 2$ (Q8/Q10) gives $g^{\mu\nu} = \mathrm{diag}(-2, 2, 2, 2) \propto \eta^{\mu\nu}$.
Closure of Q5b-O3: the open problem O3 of Q5b is resolved: the temporal coefficient is $A_\tau = 2$, derived without additional hypotheses beyond those already established.

The derivation chain and Q5b-O3

Q11 sits at the end of a long derivation chain linking the spectral admissibility programme to the emergent geometry:

Q5a → Q5b → Q7 → Q8 → Q9 → Q10 → U1 → W1 → H2 → Q11 → Q5b-O3 closed.

Each step contributes a specific structural argument:

Q5a/Q5b: formulate the admissibility hypotheses and open problems.
Q7: bridge from spectral data to co-metric coefficient equations.
Q8/Q10: identify the spatial coefficients $A_Z = A_H = 2$.
U1/W1/H2: prove the foundational hypotheses [U], [H-w].
Q11: identify the temporal coefficient $A_\tau = 2$ and close O3.

Significance. The Minkowski co-metric is not an input assumption of the Cosmochrony programme but a derived output: it emerges from the spectral structure of Heisenberg–Weil representations on admissible graphs.

Relation to the Cosmochrony programme

The full identification $g^{\mu\nu} \propto \eta^{\mu\nu}$ established by Q11 is a prerequisite for the papers that build the dynamical theory on the emergent geometry:

Q12: uses the co-metric as input for the vertical variation of the projective spectral entropy, deriving the Yang–Mills equations.
Gravity paper: uses the metric for the horizontal variation, recovering the Einstein equations.

The temporal sign (the $-2$ entry in $g^{\mu\nu}$) encodes the Lorentzian signature and traces back to the opposite orientation of the temporal BFS shells relative to the spatial shells — a consequence of the non-commutative structure of the Heisenberg group.

References

Jérôme Beau. Temporal Casimir Rigidity and Closure of the Effective Metric: Identification of A_τ = 2, 2026. doi:10.5281/zenodo.20098387